Triangles Made by Vector Addition

Place two vectors tail to tail and draw the resultant that closes the gap from the start of the first to the end of the second. You now have three sides of a triangle: the two original magnitudes and the resultant. The angle between the two original vectors sits inside that triangle, opposite the resultant side, so any tool that solves a triangle from two sides and the included angle will hand you the resultant's length. That tool is the Law of Cosines.

For vectors of magnitude a and b meeting at angle theta, the resultant magnitude is R = √(a² + b² + 2ab cos(theta)). Notice the plus sign: this looks like the Law of Cosines you may have seen written with a minus, but here theta is the angle between the vectors as placed tail to tail, which is the supplement of the triangle's interior angle, and that flips the sign. The formula is really just c² = a² + b² - 2ab cos(C) in disguise, so trust the triangle picture over memorizing a sign.

This topic connects directly to the vector addition you already know: components let you add vectors algebraically, while this triangle approach gives you the same magnitude using angles and lengths alone, no coordinates needed. It also sets up a natural question about direction, not just size, since the Law of Sines can later tell you which way the resultant points. Watching R(theta) as theta changes previews how a function's behavior at extreme inputs reveals the geometry behind it.